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A search without examples

The highest transition temperature of any equilibrium phase at ambient pressure has belonged to the mercury cuprates since 1993, at 133 to 135 K in HgBa₂Ca₂Cu₃O₈₊δ [1, 2] and 138 K with partial thallium substitution [3]. For a phonon-mediated superconductor at ambient pressure it is 39 K, in MgB₂, found in 2001 and still the record in a 2025 survey [4, 5]. Under pressure the accepted record is about 250 K, in LaH₁₀ near 170 GPa, with zero resistance, an isotope effect and reproduction by a second group [6, 7].

Later reports exceed these values, and we found no independent reproduction of any. A 2025 preprint reports resistive onsets up to 298 K in LaSc₂H₂₄ at 260 GPa, without a measurement of magnetic screening or an isotope substitution [8]. A second group saw no transition between 245 and 300 K in the four cells that gave transport data, and identified the claimed phase by diffraction in none [9]. A 2026 paper reports a pressure-quenched state of HgBa₂Ca₂Cu₃O₈₊δ with a resistive onset of up to 151 K at ambient pressure. Zero resistance is not reported [10]. Another 2026 paper claims phonon-mediated transitions at 112 to 187 K at ambient pressure in boron-doped quenched carbon [11]. Figure 1 shows both records against time.

A model built to find a superconductor at 300 K and one atmosphere therefore has no example to learn from, and the nearest verified material is a factor of 2.2 to 2.3 below the target.

A random-forest model of $T_c$ from chemical composition reaches $R^2$ of 0.88 for $\ln T_c$ on a random train-test split of the SuperCon entries above 10 K, but one trained on low-$T_c$ compounds alone underestimates cuprates and iron-based materials [12]. Random splits overstate what such a model can do in discovery [13]. SuperCon holds many doping series [12], and near-duplicate compositions account for part of the high score of a nearest-neighbour model on such a split [14].

Five superconductors first suggested by a learned model have been made with an identified phase in three peer-reviewed reports, at 0.8 to 5.4 K [15, 16, 17], and a fourth adds Pd₂NiTe₂ at 1.1 K [18, 19]. Nine alloys from a generative model superconduct at 4.8 to 9.7 K without forming the predicted structure [20], and a signal near 9 K has no isolated phase [21]. All were found at ambient pressure and are below 10 K. Mg₂RhH₆, from a machine-learning-accelerated search [22], reaches 29 K at 53 GPa [23]. The target is ten times that.

Two of the five came from a model of the electron-phonon spectral function. Its authors state that it is unlikely to predict hydride-like high-$T_c$ materials, because its training spectra are at ambient pressure and truncated at 100 meV [16].

This paper asks what limits superconductivity at 300 K and one atmosphere and what a model aimed at that target should estimate, and it makes five claims.

A transition at 300 K and one atmosphere needs a pairing scale and a phase-stiffness scale above 300 K [24] in a phase that persists at one atmosphere. LaH₁₀ at 170 GPa pairs to 250 K, has a stiffness scale far above that, and does not exist at one atmosphere. The mercury cuprate exists there with a stiffness scale of 130 to 190 K and a spectroscopic gap scale above 300 K that may not be a pairing gap [25, 26]. MgB₂ exists there with a stiffness scale of 1400 K and pairs at 39 K. No material with a reproduced transition is within a factor of two of the target on all three at once.

For phonon-mediated pairing we use a known limit, $k_BT_c\to0.1827\,\hbar\sqrt{S}$ at large coupling, where the Hopfield sum $S$ is the sum over atoms of the Hopfield parameter divided by the mass [27, 28, 29]. $S$ is an upper scale for $T_c$, and the published hydride spectra for which we solved the Eliashberg equations reach about half of it or less. By a sum-rule identity $S$ is fixed at fixed structure and linear coupling, so renormalising the phonons cannot change it. It is not protected against Fermi-surface sampling at a density-of-states peak, a change of structure under nuclear quantum motion, or a non-linear or averaged vertex.

The Hopfield parameter of hydrogen, $\eta_H$, is its number density times a scattering strength per proton, $h$. For a proton in an electron gas $h$ is the free-electron limit of the Gaspari-Gyorffy formula [30], with self-consistent calculations since 1974 [31, 32, 33, 34], and our evaluation gives about 2.3 to 2.5 $e^2$ at metallic densities. Our addition is the normalisation, with a census of 75 published calculations of 68 hydrides in which $S$ scales with hydrogen density and $h$ is 23 eV Å to within a factor of about 1.5. Megabar hydrides have 5.7 times the hydrogen density of ambient-pressure ones and an $h$ about 40% higher, so density carries most of the difference in $\eta_H$. Within each of the five megabar hydrides computed by Quan and coauthors, further pressure lowers $T_c$ [28]. At a hydrogen density of 0.10 Å⁻³, about the highest known near one atmosphere, typical values of $h$ and of the efficiency give about 70 to 150 K, and the largest $h$ and best efficiency computed so far give about 200 K. None of the factors is bounded.

In the 8,323 compounds within 50 meV per atom of the convex hull that we recovered from the figures of the largest ambient-pressure survey [5], the fraction above a given calculated $T_c$ falls exponentially between 2 and about 30 K, with a scale of 4.0 K. The sample is not a random draw, and the exponential form fails at its top, where it gives 34 K for the largest value and the sample has 54 K. These data cannot say whether a calculated 300 K lies in the tail; our conclusion that it does not rests on the Hopfield sum. If true values have an exponential tail and calculations a log-normal error whose size we have not estimated, selection on the calculated value overstates the leaders by factors we tabulate.

No hydride predicted to superconduct above 60 K at ambient pressure has been made at, or recovered to, one atmosphere. Of eleven named predictions, Mg₂IrH₆ and Mg₂PtH₆ did not form when their metals were heated in hydrogen [35, 36], Li₂AuH₆ and Li₂AgH₆ are unstable in one path-integral simulation [37], Li₂CuH₆ stayed intact in a short simulation with classical nuclei [38], and for six we found no test. Mg₂RhH₆, predicted at 45 to 59 K, is the one member of its family that has been made, and it reverted below about 30 GPa on decompression [23].

We then specify a model, roomtsc, whose supervised label is $S$, with its baselines and its tests. None of it has been trained.

Three conditions

Superconductivity at a temperature $T$ requires that electrons are paired at $T$ and that the pairs are phase coherent at $T$. Emery and Kivelson wrote this as two temperature scales, each of which limits $T_c$ [24, 25]:

$T_{\text{pair}}$ is the temperature at which pairs form, estimated from the zero-temperature gap as $k_BT_{\text{pair}}=\Delta_0/2$. $T_\theta$ is the temperature at which phase order would be lost if the pairs survived. It is set by the superfluid stiffness at zero temperature, which is obtained from the magnetic penetration depth $\lambda_L$:

with $A \approx 0.9$ and $a$ the layer spacing $d$ for a layered material, and $A \approx 2.2$ and $a=\sqrt{\pi}\,\xi$ for an isotropic one with coherence length $\xi$. Carlson and coauthors present the numerical factors as estimates that depend on microscopic details [25]. Neither inequality is a theorem. Weak-coupling BCS theory has $k_BT_c=\Delta_0/1.76$, which is above $\Delta_0/2$, and explicit models exist in which $T_c$ exceeds any fixed multiple of the zero-temperature stiffness [39]. A rigorous bound exists in two dimensions, $k_BT_c \le \pi\tilde D/2$ with $\tilde D$ fixed by the optical sum rule, which becomes $k_BT_c\le E_F/8$ for a single parabolic band [40]. We use equation (1) as an organising estimate.

The third condition is that the phase carrying the condensate exists at one atmosphere and at the operating temperature for as long as it is used, either as the thermodynamic ground state or behind a kinetic barrier. We call this persistence.

A transition at 300 K and one atmosphere therefore needs $T_{\text{pair}}$ and $T_\theta$ above 300 K in a phase that persists there. Table 1 sets three well-characterised superconductors against each condition.

No material with a reproduced transition is within a factor of two of 300 K on all three conditions at once. The highest established $T_c$ at one atmosphere, 133 to 135 K in the mercury cuprate and 138 K with thallium substitution [3], is a factor of 2.2 to 2.3 below. The pressure-quenched state of the unsubstituted compound has a resistive onset up to 151 K in one group's measurements, with no zero resistance reported, and one sample's onset fell from 147 to 143 K after cycling to room temperature [10].

LaH₁₀ pairs at 250 K, a factor of 1.2 below the target. Its cubic phase distorts below 135 GPa [42], and we found no report of LaH₁₀ at one atmosphere. Its stiffness scale, on the penetration depth inferred from magnetisation, is about a hundred times the target or more. MgB₂ exists at one atmosphere with a stiffness scale of 1400 K and pairs at 39 K, a factor of 7.7 below the target. The mercury cuprate exists at one atmosphere, and its stiffness scale of 130 to 190 K is a factor of 1.6 to 2.3 below the target. Its gap scale of 435 K, or $2\Delta_0/k_BT_c\approx13$, is above the target. Carlson and coauthors tabulate it as the pairing scale [25], which it is only if the gap seen by spectroscopy is a pairing gap, and that reading is contested. Kondo and coauthors find in Bi-2201 and Bi-2212 that pairing sets in at 120 to 150 K and attribute the pseudogap above that temperature to a state without pair formation [26]. If the same holds in the mercury compound, its pairing scale is a factor of 2.0 to 2.5 below the target.

The stiffness scale of the hydrides is inferred through models, and we found no direct measurement of the penetration depth in a hydride superconductor above 150 K. For H₃S the values derived from one laboratory's magnetisation data run from 22 to 189 nm [41, 43, 44], over which equation (2) gives $T_\theta/T_c$ from 475 down to 7.5. An October 2026 addendum to the first of these re-derives 24 nm for H₃S, which lowers the upper ratio to about 400, and 36 nm for LaH₁₀, just outside the range used in Table 1 [45]. Critical-current analyses by a second group give 211 nm for YH₆ and 500 nm for ThH₁₀ [46, 47], for which we obtain $T_\theta/T_c$ of 4.8 and 1.8. Each of these estimates, and the one for LaH₁₀, puts the stiffness scale above $T_c$, by a factor between 1.8 and about 800.

The three conditions involve different physics and are constrained by different data, so we treat them as separate estimation problems. A single regression on $T_c$ mixes them. The next three sections concern the pairing scale when phonons provide it, where a forward theory exists. The two after them concern persistence at one atmosphere, and the pairing and stiffness scales when phonons do not provide the pairing, where no controlled, validated forward theory exists.

Usability

A superconductor carries loss-free current only below its irreversibility field, which falls to zero as the temperature approaches $T_c$ and is smaller the stronger the thermal fluctuations. Gurevich gives the scale as [48]

where the Ginzburg number $Gi=\tfrac12\left[2\pi\mu_0k_BT_c\gamma\lambda_L^2/(\Phi_0^2\xi)\right]^2$ measures the strength of thermal fluctuations, with $\gamma$ the anisotropy and $\Phi_0$ the flux quantum [49]. A material with $T_c$ of 300 K therefore carries no useful current at 300 K. Pickett puts the $T_c$ needed for room-temperature use at 375 to 400 K, and more for high current density [50].

The margin needed depends on $Gi$, which is $2\times10^{-9}$ in niobium, $8\times10^{-7}$ in Nb-Ti and about $10^{-2}$ in YBa₂Cu₃O₇ [49]. Holding the coherence length, penetration depth and anisotropy of YBa₂Cu₃O₇ fixed and raising $T_c$ to 400 K gives $Gi\approx0.2$. The lower limit on the vortex creep rate proposed by Eley and coauthors, $Gi^{1/2}\,T/T_c$, is then above 0.3 at 300 K, about four times its value for YBa₂Cu₃O₇ at 77 K [49]. That limit was tested at $T_c/4$ and we extrapolate it to $0.75\,T_c$. On this estimate a layered, low-carrier-density superconductor of that kind with $T_c$ of 400 K would not hold a useful current at room temperature.

$Gi$ varies as $\lambda_L^4$, so for the hydrides it is less certain than the penetration depth. Over the 22 to 189 nm published for H₃S it runs from about $10^{-6}$ to $3\times10^{-3}$, which is four orders of magnitude below YBa₂Cu₃O₇ at one end and a third of it at the other. The critical-current analyses of YH₆ and ThH₁₀ give $3\times10^{-3}$ to $7\times10^{-3}$ and 0.039 to 0.085 [46, 47]. A free-electron estimate from the carrier density of H₃S, without mass enhancement, gives 18 nm [41], at the low end. Carrier density therefore predicts a small $Gi$ for dense three-dimensional metals of this kind, and measurement has not yet established it.

We keep usability out of the search objective and report an uncalibrated estimate of $Gi$ for every phonon-mediated candidate, from its penetration depth, coherence length and anisotropy.

What 300 K costs when phonons do the pairing

Phonon-mediated superconductivity has a forward model, the Migdal-Eliashberg equations, which return $T_c$ from the Eliashberg spectral function $\alpha^2F(\omega)$ and a Coulomb pseudopotential $\mu^*$. Fully first-principles calculations, benchmarked with density functional theory for superconductors, come within about 20% of the measured $T_c$ for most elemental superconductors [51, 52]. We use three moments of the spectral function:

To find what a 300 K transition requires we solved the linearised isotropic Eliashberg equations on the Matsubara axis for a single Einstein mode of frequency $\omega_E$ (methods and checks are in the appendix). For this spectrum $k_BT_c = \hbar\omega_E\,f(\lambda,\mu^*)$, and Table 2 gives $f$ and the frequency that 300 K requires at each coupling.

At $\lambda = 1$ the required frequency is 357 meV. Trachenko and coauthors estimate from fundamental constants that phonon frequencies in a non-molecular solid stay below 317 meV (3680 K), which a single mode would meet at $\lambda = 1.1$; some calculated spectra under pressure exceed the estimate [53]. At the second-moment frequencies of hydrogen in the megabar hydrides of Table 3, 113 to 163 meV [28], a single mode needs $\lambda$ between 1.9 and 3.1. The table assumes an isotropic gap, a constant density of states and Migdal's approximation, whose expansion parameter $\lambda\hbar\omega/E_F$ [29] is below 0.1 in these rows only where the Fermi energy exceeds 3 to 4 eV.

Figure 2 places published calculations against the same solution. Among the 12,053 compounds in the coupling figure of the largest ambient-pressure survey, which reports more than 20,000, none combines $\lambda$ above 2 with $\omega_{\log}$ above 600 K, and the largest $\omega_{\log}$ is 1275 K, at $\lambda = 0.23$ [5]. The abstract of that paper quotes 1800 K; the appendix explains why we use the figures.

Which phonons count

$T_c$ depends on the frequencies at which the coupling sits, and the total $\lambda$ does not record them. We recomputed the functional derivative $\delta T_c/\delta\alpha^2F(\Omega)$ of Bergmann and Rainer, the response of $T_c$ to a small spectral weight added at frequency $\Omega$ [54], for an Einstein spectrum with $\lambda = 2$. It peaks near $\hbar\Omega = 7\,k_BT_c$. Per unit of spectral area, the most valuable phonons for a 300 K superconductor are therefore near 180 meV, in the range of hydrogen stretching vibrations.

Per unit of $S=\lambda\langle\omega^2\rangle$, which the next subsection calls the Hopfield sum, the ranking reverses. A weight $\epsilon$ added at $\Omega$ adds $2\Omega\epsilon$ to $S$, so the response per unit of $S$ is the same derivative divided by $2\Omega$. It decreases monotonically with frequency: relative to its low-frequency value it is 0.92 at $2\,k_BT_c$, 0.56 at $6\,k_BT_c$ and 0.22 at $13\,k_BT_c$. At fixed $S$ a softer spectrum gives a higher $T_c$. The gain is limited, because the efficiency $\Phi$ of equation (6) is below one in every solution we computed and is already 0.63 for a single mode at $\lambda=2$, and because a mode can soften only until the lattice becomes unstable. This section treats $S$ as the scarce quantity, so the second ranking applies.

Split $\lambda = 2$ between a 15 meV mode, typical of a heavy host lattice, and a 150 meV mode, typical of hydrogen. With all of the coupling on the soft mode $T_c$ is 30 K, with an even split 134 K, and with all of it on the stiff mode 285 K. This comparison is at fixed $\lambda$. The Hopfield sums of the two extremes differ by a factor of 100 and their $T_c$ by about ten, its square root.

The Hopfield sum

One moment of the spectral function does not depend on the phonon frequencies. McMillan showed for a metal with one kind of atom that $\lambda\langle\omega^2\rangle = N(0)\langle I^2\rangle/M$, and Hopfield identified the numerator as a local electronic quantity [55, 56]. With one term for each atom type, as it is written for hydrides [28, 57], the relation is

where the sum runs over atom types, $M_j$ is the mass, $N(0)$ the density of states per spin at the Fermi level and $\langle I_j^2\rangle$ the Fermi-surface average of the squared electron-ion matrix element, summed over the atoms of type $j$ in the cell. $\eta_j$ has the units of a force constant. The phonon frequencies and eigenvectors cancel out of $S$ because the eigenvectors form a complete set. We call $S$ the Hopfield sum and quote it in the units of $\eta_H$ (eV/Ų) by multiplying by the hydrogen mass. For the megabar hydrides of Table 3 it equals $\eta_H$ to within 3%. Where lithium, beryllium or boron couple, or hydrogen is a small part of the cell, it exceeds $\eta_H$ by an amount we have not measured (appendix).

Allen and Dynes found that at large coupling the Eliashberg equations give $k_BT_c\to0.18\,\hbar\sqrt{\lambda\langle\omega^2\rangle}$, with a constant of 0.1827 at $\mu^*=0$, and concluded that the phonon frequencies do not limit $T_c$ [27, 29]. By equation (5) the limit is $0.18\,\hbar\sqrt{\eta/M}$ for one atom type, as Quan, Ghosh and Pickett and Sadovskii write it [28, 29]. We use this known limit as a scale and write, at any coupling,

where the efficiency $\Phi$ depends on $\lambda$, on $\mu^*$ and on the shape of the spectrum. For hydrogen the prefactor is $136.5\ \text{K}\times\sqrt{\eta_H/(\text{eV Å}^{-2})}$. $\Phi$ is below one in every isotropic solution we have computed, and the bound of this form proved for a general spectrum has a constant 2.03 times larger [58]. $S$ is therefore an upper scale for $T_c$. Our additions are $\Phi$ for single modes and for published spectra, $S$ and $\eta_H$ for ambient-pressure hydrides from published tables, and an account of which errors $S$ is protected against.

For a single mode with the $\mu^*$ of Table 2, $\Phi$ is 0.40, 0.55, 0.63, 0.71, 0.75 and 0.81 at $\lambda$ = 1, 1.5, 2, 3, 4 and 10. Our solutions for four published spectra of hydrides predicted at ambient pressure give 0.35 to 0.52, against 0.50 to 0.68 for a single mode with the same $\lambda$ and $S$, and 0.03 for the anharmonic spectrum of PdH, where $\lambda = 0.40$. With the first-principles Coulomb interaction of the authors of three of those spectra, the published $T_c$ is 0.21 to 0.41 of the asymptote [22]. For the five superhydrides of Table 3 the hydrogen-mode $T_c$ of Quan and coauthors, an Allen-Dynes value that we reproduce with $\mu^* = 0.13$, a stronger Coulomb term than that of Table 2, is 0.51 to 0.58 of the asymptote [28]. Leaving PdH aside, $\Phi$ runs from about 0.2 to 0.6.

Once the structure and the linear electron-ion matrix elements are fixed, equation (5) fixes $S$ whatever the frequencies and eigenvectors are, so renormalising the phonons at a fixed structure cannot change it. The published anharmonic calculations change only the phonons, and their tables obey the rule to rounding. For Li₂AuH₆ with harmonic phonons and two anharmonic treatments, $\lambda$ is 3.86, 2.10 and 2.67, and the $S$ we compute from the tabulated $\lambda$ and second moment is 3.62, 3.63 and 3.62 eV/Ų [5]. For PdH $\lambda$ falls from 1.55 to 0.40, and our reading of the published cumulative coupling curves gives 0.42 eV/Ų for both, within a reading error of about 15% and with the harmonic calculation done at the volume of PdT [59]. This agreement is an identity. It checks the tables and is no evidence about the accuracy of $S$.

The sum rule does not protect $S$ against errors in the electronic factor or in the structure, and three kinds are documented in hydrides. Three published calculations of one structure of Mg₂IrH₆, whose Fermi level sits on a peak in the density of states, give 2.3, 3.7 and 5.2 eV/Ų (Table 3). Path-integral dynamics pairs part of the hydrogen in Li₂AuH₆ into molecules, and $S$ in that state is 0.86 eV/Ų against 3.6 for the crystal, a factor of 4.2 between two different methods [5, 37]. In PdH, on the same anharmonic phonons, $\lambda$ is 0.42 with the bare linear vertex, 0.27 when that vertex is averaged over the quantum distribution of the nuclei, and 0.64 when averaged second-order terms are added [60].

At fixed $S$ an error in the phonons moves $T_c$ less than the same fractional error in $\eta$. With $a = d\ln T_c/d\ln\lambda$ at fixed $S$, errors in $\eta$ and in the force constants $k$ give $\delta T_c/T_c = (\tfrac12 + a)\,\delta\eta/\eta - a\,\delta k/k$. For a single mode $a$ is 0.61, 0.39, 0.22 and 0.15 at $\lambda$ = 1.5, 2, 3 and 4. Real spectra are more sensitive than a single mode with the same $\lambda$, because their soft modes add to $\lambda$ and little to $S$ or $T_c$. Scaling every force constant in the four ambient-pressure spectra by 1.3 or 1/1.3 at fixed $S$ moves $T_c$ by 12 to 25%, and the same factor in $\eta$ at fixed frequencies moves it by 24 to 41%, about twice as much. A non-uniform change in the phonons can exceed the single-mode rule. In H₃S at 200 GPa anharmonicity lowers $\lambda$ from 2.64 to 1.84 and the Eliashberg $T_c$ from 250 to 194 K, where the rule gives 222 K [61]. At small $\lambda$ a fixed $S$ constrains $T_c$ little, and in PdH it falls from 47 to 5 K [59].

In these terms the requirement for 300 K is the last column of Table 2: $\eta_H$ of 12.0 eV/Ų at $\lambda = 2$ and 8.7 at $\lambda = 4$ for a single mode, with a floor of 4.83 at infinite coupling and $\mu^*=0$. At the efficiencies of the megabar hydrides (0.51 to 0.58) it is 14 to 19 eV/Ų, and at those of the four ambient-pressure spectra (0.35 to 0.52) it is 18 to 39. The floor scales with mass, so boron needs at least 52 eV/Ų and carbon 58, about four times the largest value in Table 3. Semenok, Altshuler and Yuzbashyan conclude from a different argument that phonon-mediated room-temperature superconductivity is feasible only in hydrogen compounds [62].

Table 3 compares the requirement with published calculations. We found no paper that prints $\eta_H$ for the hydrides predicted at ambient pressure, but $\lambda$ and the second moment are tabulated for more than sixty of them and four published spectra cover three, so the values follow from equation (5).

The five megabar hydrides of the table, each with a calculated $T_c$ above 200 K, have 6.7 to 13.5 eV/Ų, at 150 to 300 GPa. The hypothetical ambient-pressure compounds have 1.6 to 5.2, a factor of 1.7 to 7.5 short of the single-mode requirement. PdH, the one ambient-pressure hydride in the table that exists, is short by a factor of more than twenty.

The three rows for Mg₂IrH₆ are calculations of one structure, and their published $T_c$ comes from different methods: the Allen-Dynes formula with $\mu^* = 0.10$ (59 K), the Eliashberg equations and density functional theory for superconductors with a first-principles Coulomb interaction (77 K and 66 to 71 K), and anisotropic Eliashberg equations with $\mu^*$ of 0.10 to 0.16 (160 to 175 K). At a common $\mu^*$ of 0.10 our solutions of the two published spectra give 134 and 174 K.

Within one phase, compression raises $\eta_H$ and lowers $T_c$, because the force constants stiffen faster. In the calculations of Quan and coauthors, between 150 and 300 GPa $\eta_H$ of CaH₆ rises from 6.7 to 8.8 eV/Ų, $M_H\langle\omega^2\rangle$ rises by a factor of 2.1, and the $T_c$ of the full spectrum falls from 200 to 175 K. Their other four compounds behave in the same way [28].

The ambient-pressure candidates have low efficiency for the reason given under "Which phonons count". In our reading of the published spectra of Mg₂IrH₆ and Mg₂RhH₆, modes above 150 meV hold about 60% of $S$ and contribute $\lambda$ of 0.2 to 0.4 [22, 64]. In Li₂AuH₆ about 70% of $\lambda$ comes from modes below 30 meV [68].

Proposed ceilings

None of the proposed bounds on $T_c$ for phonon pairing reviewed below excludes 300 K, and they disagree about the range of coupling in which a room-temperature material would have to lie.

Esterlis, Kivelson and Scalapino proposed $k_BT_c \lesssim 0.1\,\hbar\bar\omega$ from measured ratios of $T_c$ to the Debye temperature and from quantum Monte Carlo for the two-dimensional Holstein model, from which they infer a maximum near 0.08 of the bare phonon frequency and 0.12 of the renormalised one [69]. In that model the simulations agree with Migdal-Eliashberg theory up to a bare coupling of about 0.4 and depart from it beyond [70]. In the same calculations bare couplings of 0.4 and 0.5 correspond to renormalised $\lambda$ of 1.7 and 4.6, a range that contains the four superhydrides of Figure 2 [69, 71]. H₃S sits at 0.08 if $\bar\omega$ is the top of its calculated phonon spectrum, 0.23 eV [69]. Kivelson and other coauthors later built models in which the ratio is unbounded, and concluded that the bound is no theorem and is hard to exceed in real materials [39].

Semenok, Altshuler and Yuzbashyan argue that an instability of the electron-phonon system limits $\lambda$ to 3.7 for Einstein and 4.7 for Debye spectra [62]. Zhang, Berg and Chubukov dispute the instability and the proponents have replied [72, 73]. Sadovskii's review agrees with Zhang and coauthors on the equilibrium specific heat and leaves the kinetic argument of the reply open [29]. He argues that the renormalised $\lambda$ is unbounded inside the stable metallic phase, takes equation (6) with $\Phi = 1$ as the upper limit on $T_c$, and estimates it for an electron gas as $(0.2/r_s)\sqrt{m/M}$ Ry, about 650 K for hydrogen at $r_s = 1$ [29]. Trachenko and coauthors bound the phonon frequency by fundamental constants and obtain a maximum $T_c$ between 370 and 740 K [53].

Leavens showed that $k_BT_c\le0.2309\,A$ at $\mu^*=0$, with $A$ the area under $\alpha^2F$ [29, 74]. Our solver reproduces the constant for a single mode at $\lambda = 1.14$ and gives 0.166 with the $\mu^*$ of Table 2, so a single mode needs $A$ of at least 156 meV for 300 K. Moussa and Cohen derived two further bounds within Eliashberg theory [75].

A cap on $\lambda$ at 3.7 lowers the single-mode $\Phi$ from 0.81, its value at $\lambda = 10$, to 0.74. A cap on $k_BT_c/\hbar\bar\omega$ of 0.10 to 0.12, with $\bar\omega$ the top of the spectrum, removes every row of Table 2 from $\lambda = 1.5$ up, because for a single mode the top of the spectrum is $\omega_E$. A single mode then needs $\omega_E$ of 2500 to 3000 K at $\lambda$ of 1.25 to 1.46, which is $\eta_H$ of 16 to 20 eV/Ų against 8.7 to 12.0 without the cap, above every value in Table 3. At 0.08, where H₃S sits, 300 K needs a top frequency of 3750 K, above the 3680 K estimate of Trachenko and coauthors. The cap does not hold on a mean frequency, since LaH₁₀ superconducts at 250 K with a calculated $\omega_{\log}$ of 1119 K, a ratio of 0.22 [6, 76]. Whether it holds on the top frequency of a three-dimensional multiband hydride is not settled. Under either cap the requirement falls on $S$, and the section "What hydrogen density buys at one atmosphere" asks what sets it.

How wrong a computed $T_c$ can be

Any model trained on computed electron-phonon data inherits the errors of the calculations. For strongly coupled hydrides the documented cases run from 7% to a factor of three between published values for one structure, and to a factor of four or more where quantum nuclear motion changes the structure (Table 4).

Anharmonic calculations come within about 10% of the measured $T_c$ of H₃S and LaH₁₀ [61, 76]. The high-throughput calculations, which make up most of the published data, are harmonic, use $\mu^* = 0.10$ and run on coarse grids [5, 63]. H₃S was predicted at 191 to 204 K at 200 GPa and measured at 203 K at 155 GPa [66, 79]. YH₆ was measured at 224 K against a prediction above 270 K and YH₉ about 30 K under its predictions, and the predicted YH₁₀ did not form [65, 80]. Mg₂RhH₆ has onsets of 18 to 29 K at 30 to 74 GPa against about 60 K calculated at 30 to 50 GPa [23]. Ranking on a calculated value adds a further overestimate of the leaders, which the subsection "Selection on the calculated value" calculates under stated assumptions.

What hydrogen density buys at one atmosphere

We define the scattering strength per proton, $h$, by dividing the Hopfield parameter of hydrogen by the hydrogen number density,

where $n_H$ is the number of hydrogen atoms in a cell of volume $V$, $\nu_F$ is the density of states per spin per unit volume, and $s^2 = V^2\langle I_H^2\rangle/n_H$ is the squared matrix element of the force on one proton for states normalised to unit volume. $h$ has the units of energy times length and does not depend on the choice of cell. Equation (7) is a definition and assumes nothing. It is useful to the extent that $h$ varies less between compounds than $\eta_H$ does, which the rest of this section measures.

A proton in an electron gas

For one proton in a uniform electron gas $h$ is a known quantity,

where the $\delta_l$ are the phase shifts at the Fermi momentum $k_F$ and the sum is the transport cross-section of the proton in units of $4\pi/k_F^2$. This is the free-electron limit of the formula of Gaspari and Gyorffy [30], which Papaconstantopoulos and Klein applied to metallic hydrogen in 1977 [81]. The same cross-section gives the residual resistivity of a dissolved proton and the friction coefficient of a slow proton, $Q = mk_Fh/3\pi\hbar$. The friction was calculated from self-consistent phase shifts in 1981 [82], and such values agree with the measured stopping of slow ions better than linear theory does [83, 84]. Self-consistent density-functional calculations of a proton in an electron gas date from 1974 [31, 32, 33], and Fermi-level phase shifts and transport cross-sections of embedded atoms were tabulated in 1983 [34].

Table 5 gives our own evaluation, a self-consistent Kohn-Sham calculation in the local-density approximation (appendix). At $r_s = 2.5$ it returns $\delta_0 = 1.217$ and $\delta_1 = 0.091$, against 1.221 and 0.089 published by Nagy and Zawadowski [84].

For $r_s$ between 1.5 and 3, $h$ is 2.3 to 2.5 $e^2$ (33 to 37 eV Å), which we use as the single-proton reference. The Yukawa model of the last column, which has been used for ion stopping and for a proton in an electron gas [85, 86], satisfies the same Friedel sum rule and is 17 to 21% lower for $r_s$ up to 3.

With equation (6), equation (7) gives

where $\Omega_p$ is the plasma frequency of the protons. In practical units the asymptote, at $\Phi = 1$, is $136.5\ \text{K}\times\sqrt{\rho_H h}$ with $\rho_H$ in Å⁻³ and $h$ in eV Å. For $h = 2.4e^2$ the asymptote is $0.080\,\hbar\Omega_p$, an estimate for hydrogen in a free-electron-like metal. It does not bound $T_c$, since $h$ in H₃S is about 3 $e^2$ (Table 6). Sadovskii gives a dimensional estimate of the same asymptote for one electron per atom, $(0.2/r_s)\sqrt{m/M}$ Ry with factors of order unity dropped, and quotes about 650 K at $r_s = 1$ [29]. Equation (9) with the $h$ of Table 5 and one proton per electron gives 880, 369 and 114 K at $r_s$ = 1, 2 and 4. His estimate scales with density as $r_s^{-1}$ and ours as $r_s^{-3/2}\sqrt{h}$.

Equation (9) is McMillan's expression for $\lambda\langle\omega^2\rangle$ in simple metals, with the phase-shift sum in place of his average of the pseudopotential form factor [55], joined to the asymptote of Allen and Dynes [27]. In simple metals the Hopfield parameter is known empirically to scale as the inverse volume under pressure [87], which makes $h$ constant within a metal. Quan and coauthors normalised the hydrogen coupling by the density of states and found that the squared matrix element differs by a factor of five among the five superhydrides [28]. We add to these known results the division of $\eta_H$ by the hydrogen density of a compound, taken separately from its electron density, and the comparison of $h$ across pressures and compounds that follows. We have not found either in the literature.

The single-site assumption

Equation (8) is exact, within the Kohn-Sham description, for one proton in a uniform gas. Applying it to each proton of a crystal assumes that the protons scatter free-electron states independently. That is not controlled in the superhydrides, where the mean proton spacing is 0.5 to 0.7 Fermi wavelengths. In a compound the formula of Gaspari and Gyorffy weights each term of the sum by the site-projected densities of states of the two channels relative to those of a single scatterer [30], and these band factors can lie on either side of one. Published rigid-muffin-tin values of $\eta_H$ for H₃S, LaH₁₀ and YH₁₀ [57, 88] are 1.2 to 1.9 times smaller than the linear-response values at the same or a nearby pressure [28]. The comparison is ours and assumes that Hutcheon and coauthors sum $\eta_H$ over the ten hydrogen atoms; Papaconstantopoulos and coauthors report a $\lambda$ consistent with linear response. We therefore take every value of $h$ for a compound from linear-response calculations, and use the single-proton value as a reference scale.

Computed hydrides

Table 6 lists $h$ for the hydrides of Table 3. The paper that reports the megabar $\eta_H$ prints no cell volumes [28], so each megabar density uses a volume from another calculation of the same structure [61, 89, 90, 91, 92, 93]. Five of the eleven volumes behind the megabar rows are printed at the pressure used, and the others are interpolated, extrapolated or read from a figure. Where two calculations print a volume at the same pressure they differ by about 2%.

Within each superhydride $h$ is constant to about 3% under pressure, so $\eta_H$ follows the hydrogen density. Over the pressure ranges tabulated by Quan and coauthors, which reach 400 GPa for MgH₆ and YH₁₀, $\eta_H$ rises by 3 to 31% and the $T_c$ they calculate from the hydrogen modes falls by 4 to 16%, for the reason given in the previous section [28].

The five superhydrides span 24 to 45 eV Å, a factor of 1.9 that is about as wide as the spread of their $\eta_H$, so density does not order them. The six candidates at one atmosphere span 19 to 45 eV Å, apart from one calculation of Mg₂IrH₆ at 64, at hydrogen densities three to five times lower. Against the single-proton value the superhydrides lie at 0.7 to 1.3, with a geometric mean of 31 eV Å, about 0.9. The three calculations of Mg₂IrH₆ lie at 0.8, 1.2 and 1.8, and PdH at 0.2. In PdH a tenth of the density of states at the Fermi level is on hydrogen, against nearly half in H₃S [95].

A census of published calculations

We computed $h$ for every hydride in the published tables we know of that give both $\lambda$ and the second-moment frequency $\omega_2$, where we could find the cell. That gives 62 ambient-pressure hydrides, PdH, and 12 pressure points of the five superhydrides, one of them (YH₁₀ at 400 GPa) with an estimated volume. Of the 62, 57 are from the tables of Cerqueira and coauthors, which list compounds with a harmonic Allen-Dynes $T_c$ above 20 K after a machine-learning pre-screen [63], and five are from Gao and coauthors [5]. For these rows the numerator is the Hopfield sum, $S = M_H\lambda\omega_2^2$, which also contains the coupling of the other atoms. The cells are the PBE cells of the Alexandria database [96], matched by formula and structure. The coupling was computed in PBEsol cells [5, 63], which are smaller by a median of 3% in the 24 database records we compared, so the ambient values of $h$ are high by about 3%. Table 7 gives the statistics and Figure 3 shows the rows.

The largest ambient value, 83 eV Å, is Be₈H, whose coupling is mainly on beryllium modes [63], so its $S$ is not $\eta_H$. Between the ambient and megabar groups the geometric mean of $S$ rises from 1.24 to 9.9 eV/Ų, a factor of 8.0. Hydrogen density accounts for a factor of 5.7 (0.057 to 0.32 Å⁻³) and $h$ for 1.4 (22 to 31 eV Å).

All but two of the ambient rows were selected for high calculated $T_c$, and $h$ rises with it. Its geometric mean is 18 eV Å below 30 K, 23 from 30 to 50 K and 30 above 50 K. Small values are therefore under-represented, and 22 eV Å is the mean of a selected sample. The smallest values in the census are 7.2 (PdH), 7.4 (NaNiH₃) and 9.4 (ZrH₃).

Hydrogen density at one atmosphere

Among compounds stable at one atmosphere and room temperature, the densest hydrogen we find from crystallographic cell volumes is 0.090 atoms per ų, or 150 kg per cubic metre, in the metal TiH₂ [97] and the insulator Mg₂FeH₆ [97, 98]. Liquid hydrogen has 0.042 [98]. VH₂, a metal in equilibrium with about two atmospheres of hydrogen at 25 °C [99, 100], reaches 0.103 at full stoichiometry [97], and we found no compound above it. The candidates of Table 6 have 0.081 to 0.086 and the superhydrides 0.23 to 0.41.

The first effect of pressure is to make metallic phases with hydrogen states at the Fermi level exist at all, and Table 6 does not show it. Quan and coauthors call the stabilisation of structures of atomic hydrogen the essential role of pressure [28]. The six candidates are hypothetical at one atmosphere. In the chemical systems of the five superhydrides, the hydrogen-richest phases that exist at one atmosphere (H₂S, LaH₃ and YH₃ [101], CaH₂ and MgH₂) are insulators or semiconductors, for which $\eta_H$ is zero. In the two existing metallic hydrides for which we have a computed hydrogen coupling, it is small. PdH has $h = 7.2$ eV Å, and for TiH₂ a calculation finds the hydrogen contribution to the coupling much smaller than in PdH or H₃S [102].

An estimate at the densest packing

We take a round density of 0.10 Å⁻³, between TiH₂ and VH₂. The asymptote of equation (9) is then 203 K for the census mean of 22 eV Å, 261 K for the single-proton value of 36.6, 286 K for the 44 of H₃S, and 345 K for the 64 of one calculation of Mg₂IrH₆, the largest value we have extracted from a published calculation whose coupling is on hydrogen. The hydride spectra of the previous section reach 0.35 to 0.52 of their asymptote at one atmosphere and 0.51 to 0.58 at megabar pressure, the second range with a stronger Coulomb term, and a single mode at $\lambda = 4$ reaches 0.75. With typical values, $T_c$ at this density is about 70 to 120 K for the census mean and 90 to 150 K for the single-proton value. With the largest scattering strength and the best efficiency computed so far it is about 200 K, which restates the 175 K calculation of Mg₂IrH₆ [64] at a 23% higher hydrogen density. A single mode at $\lambda = 4$ with the same $h$ would give about 260 K. If $k_BT_c$ cannot exceed 0.10 to 0.12 of the top phonon frequency, the cap discussed under "Proposed ceilings", a single mode reaches at most 0.49 to 0.54 of the asymptote, which is 170 to 190 K. These are conditional estimates. The efficiency is below one in every solution we computed and is otherwise unconstrained, and we know of no bound on $h$ or on the hydrogen density of a metastable phase. The estimates may also be high, since among the ambient rows of the census $h$ falls as hydrogen density rises, and the seven rows above 0.090 Å⁻³ have $h$ between 9 and 29 eV Å.

A 300 K transition needs $\rho_Hh$ of 8.7 to 12 eV/Ų with a single mode (Table 2). At 0.10 Å⁻³ that is a scattering strength of 87 to 120 eV Å, which is 2.4 to 3.3 times the single-proton value, 2.0 to 2.7 times the largest megabar value and 1.4 to 1.9 times the largest in Table 6. Under the cap on the top frequency the single-mode requirement is 16 to 20 eV/Ų, or 160 to 200 eV Å at this density. At megabar scattering strengths, 24 to 45 eV Å, it is a hydrogen density of 0.19 Å⁻³ or more, twice that of the densest compound we found stable at one atmosphere.

The estimates would be too low if band structure concentrates the states at the Fermi level on hydrogen, if the coupling is not linear in the hydrogen displacement, or if hydrogen packs more densely in a metastable structure than in any stable one.

In Mg₂IrH₆ the Fermi level sits on a peak in the density of states, and three calculations give $h$ from 28 to 64 eV Å depending on how the peak is sampled [35]. RbPH₃ may lie above the range of Table 6. Its $T_c$ of 126 K from the isotropic Eliashberg equations at $\mu^* = 0.14$, at 0.040 hydrogen atoms per ų [103], implies $S/\rho_H$ of at least 64 eV Å unless its efficiency exceeds 0.58, and the 90 K the same paper gives with a first-principles Coulomb interaction implies at least 33. Its Hopfield sum has not been published.

In PdH, Bianco and Errea average the vertices over the nuclear distribution, and adding the second-order vertex then raises $\lambda$ from 0.27 to 0.64. Of their four treatments only this one gives the inverse isotope effect and a $T_c$ near the measured one, 11 K against 8 to 9 K, and the second-order term with bare vertices gives 50 K [60]. If effects of that size are general, linear-response values of $S$ are low by an unknown amount.

The cubic PdH₄ predicted by Li and coauthors [104] has 0.118 hydrogen atoms per ų in its cell in the Alexandria database, where it lies 319 meV per atom above the hull [96].

MgB₂ has a scattering strength per atom several times the single-proton value, and this implies nothing for hydrogen. From $\lambda = 0.87$ and $\omega_{\log} = 725$ K [105] its Hopfield sum in boron units is at least 8.8 eV/Ų, which at a boron density of 0.069 Å⁻³ is 128 eV Å per boron atom. The electron-gas calculation of Table 5, repeated for a boron nucleus at the valence density of MgB₂ ($r_s = 1.80$), gives 121 eV Å, so MgB₂ sits at its electron-gas value.

What the census supports

Across published calculations the Hopfield sum scales with hydrogen number density. The slope of $\ln S$ on $\ln\rho_H$ over the 75 rows is $1.03\pm0.06$, with a scattering strength per proton of 23 eV Å to within a factor of about 1.5, which is 0.6 to 0.7 of the single-proton value. One common $h$ describes the step from one atmosphere to megabar pressure.

The census does not support a constant $h$ among ambient-pressure compounds. Within the 62 ambient rows the slope is $0.62\pm0.09$, the correlation of $\ln h$ with $\ln\rho_H$ is $-0.47$, and $h$ is only slightly narrower than $S$ itself, with a standard deviation of the logarithm of 0.36 against 0.42. The three calculations of Mg₂IrH₆ alone have a standard deviation of the logarithm of 0.42, so part of the spread between compounds comes from the settings of the calculations. The census does not support the same $h$ at megabar pressure as at one atmosphere either, since megabar values are about 40% higher.

Gao and coauthors deposited their electron-phonon data in the Alexandria database [5, 96], and the dynamically stable records we inspected store $\lambda$ and $\omega_2$ with the spectrum resolved by mode. We have not checked that compounds below the $T_c$ cut are among them and have not computed $h$ from them, so its distribution among unselected hydrides and the hydrogen share of $S$ in each row remain to be measured.

What screening at ambient pressure has found

The largest set of calculated $T_c$ at ambient pressure that we found is that of Gao and coauthors, who report more than 20,000 metals [5]. From the vector figures of their paper we recovered the Allen-Dynes $T_c$ (harmonic phonons, $\mu^* = 0.1$) and the energy above the convex hull of 12,079 of them. The calculation is isotropic, and its authors note that it underestimates MgB₂, measured at 39 K, by almost a factor of two [4, 5].

The compounds were not drawn at random. The core of the set is the training set of an earlier screen by the same group, 6,912 non-magnetic metals within 50 meV per atom of the hull with small high-symmetry cells and an estimated Debye temperature above 300 K [106]. To these were added the compounds that a model predicted above 5 K, hydrides predicted above 20 K, and in the later survey compounds predicted above 10 K [5, 63, 106]. We analyse the 8,323 compounds within 50 meV per atom of the hull. All passed the harmonic stability filter of the source calculations [5, 106], and the statistics below describe that sample.

An exponential tail between 2 and 30 K

The mean amount by which the calculated $T_c$ exceeds a threshold is constant within its standard error from 2 to 15 K. It is $4.10\pm0.08$ K above 2 K (2,482 compounds), $4.08\pm0.12$ above 5 K, $4.01\pm0.16$ above 7.5 K, $4.03\pm0.23$ above 10 K (355), $3.92\pm0.34$ above 12.5 K and $4.02\pm0.53$ above 15 K (100). Above 20 K, where 25 compounds remain, it is $5.05\pm1.58$ K. A constant mean excess is the property of an exponential tail, the zero-shape generalised Pareto case of the peaks-over-threshold method [107, 108],

with a bootstrap 95% interval of 3.6 to 4.5 K for $\tau$ that treats the compounds as independent. An Anderson-Darling test does not reject the exponential form above 10 K ($p = 0.59$). The source of the training set reported 214 of its 6,912 compounds above 10 K, and chances of about 0.4% above 20 K and 0.03% above 30 K [106], which correspond to scales of 4.9 and 3.9 K. The mean-excess analysis and the fitted form are ours. Between 2 and about 30 K a threshold 9.3 K higher is exceeded by one tenth as many compounds (Figure 4). The fraction $p_u$ is the composition of this sample and carries over to no other pool.

The top of the sample

If equation (10) held at all temperatures, the level exceeded once among $N$ compounds chosen the same way would be

and the largest of the $N$ would scatter about it with a standard deviation of $1.28\,\tau$, which is 5 K. For the 355 compounds above 10 K the level is 34 K. The sample has Mg₂RhH₆ at 54 K, 49.7 meV per atom above the hull, and LiMoN₂ at 42 K. Under equation (10) a largest value of 54 K or more has a probability of 0.007. The equation expects 2.5, 0.27 and 0.017 compounds at or above 30, 39 and 50 K, and the sample has 4, 2 and 1. The exponential form fails at the top of the sample it was fitted to. The failure rests on those two compounds. With the hull cut at 49 meV per atom, which removes Mg₂RhH₆, the largest value is 42 K and its probability is 0.09.

With the shape left free the data do not determine the tail. A generalised Pareto fit has a shape of 0.05 above 10 K, with a bootstrap 95% interval of −0.13 to 0.15, and 0.16 (−0.14 to 0.35) above 15 K. Return levels are therefore illustrations, conditional on the shape and on $p_u$. For a million compounds equation (11) gives 53 K if 4.3% exceed 10 K, as here, and 42 K if 0.27% do. That is the fraction confirmed among the roughly 200,000 compounds of the earlier screen, a lower bound because most were calculated only if its model flagged them [106]. With $p_u$ as in the sample, the two generalised Pareto fits give 63 K (34 to 104 K) and 89 K (35 to 230 K).

For a calculated 300 K the three fits give probabilities per compound of $2\times10^{-33}$, $4\times10^{-16}$ and $8\times10^{-10}$, so these data make no statement about 300 K either way. The argument that ranking this population will not produce a 300 K compound is the one from the Hopfield sum. A single mode with $\lambda$ between 4 and 2 needs a hydrogen Hopfield parameter of 8.7 to 12 eV/Ų for 300 K (Table 2), and the largest we have found in a published ambient-pressure calculation is 5.2 (Table 3).

One later screen gives a weak check of the scale. Among the stable cubic hydrides of the GNoME database, 25 have an Allen-Dynes $T_c$ of 4.2 K or more, and the largest is 23.5 K [109]. For 25 exceedances equation (11) gives 17 K, and a largest value of 23.5 K or more has a probability of 0.19. The candidates were chosen by a model whose training data include the 2024 screens behind the sample fitted here [63, 106], and the count of 25 is an input, so $p_u$ is not tested.

Best calculated $T_c$ against hull distance

Beyond 50 meV per atom the compounds are in the set because a model predicted a high $T_c$ for them or an earlier paper proposed them [5, 63, 106]. The share above 10 K is 24%, 43% and 51% in the ranges 50 to 100, 100 to 200 and 200 to 376 meV per atom, against 4.3% inside 50. We assign no tail scale to those ranges.

The best calculated $T_c$ at or below a given hull distance, the front drawn in the source figure [5], is a record of what has been found. It is 22.9 K on the hull, 42.0 K within 25 meV per atom, 53.8 K within 50, 59.4 K within 100, 88.4 K within 200 and 110.6 K at 319, a rise of about 0.3 K per meV per atom. Past LiMoN₂ the front runs through hydrides only, Mg₂RhH₆, Mg₂IrH₆, KInH₃, Li₂AuH₆ and Li₂AgH₆.

Selection on the calculated value

Ranking on a noisy number overstates the compounds at the top, an effect known in decision analysis as the optimizer's curse [110]. We calculate its size on two assumptions. The true $T_c$, $t$, has an exponential density of scale $\tau_t$. The calculated value is $y = t\,e^{\epsilon}$, with $\epsilon$ normally distributed with zero mean and standard deviation $\sigma$. The posterior density of $t$ given $y$ is then proportional to $\exp[-t/\tau_t-\ln^2(y/t)/2\sigma^2]$, and at its mode the ratio $R = y/t$ satisfies

This is a special case of the correction for Eddington bias in survey astronomy [111], whose prior-free form is Tweedie's formula [112], and we apply it to calculated $T_c$. The ratio grows with $y/\tau_t$ and requires no bias in the calculation.

We have not estimated $\sigma$, the spread of the high-throughput value about the true one across the screened population, and we found no published estimate. The calculated and measured values tabulated for known superconductors could supply one, and with it a test that we have not made [113]. The only number we have is the spread of six published values of $T_c$ for one compound, Mg₂IrH₆ (two are midpoints of quoted ranges) [22, 35, 63, 64, 94]. It is 0.39 in $\ln T_c$, with a 95% interval of 0.25 to 0.97, and 0.22 without the largest value. The compound is atypical, with its Fermi level on a peak in the density of states [35] and an Alexandria hull distance of 86 meV per atom [96]. The zero mean is an assumption as well, since the errors of Table 4 have signs and their net is unknown.

Whatever the distribution of true values, calculated values with log-normal error of width $\sigma$ cannot have an exponential tail of scale $\tau$, as in equation (10), above $y = \tau/\sigma^2$, where Tweedie's formula would make the posterior variance negative. The limit is 45 K at $\sigma = 0.3$ and 25 K at $\sigma = 0.4$. The simulation behind Table 8 keeps the two assumptions, and its calculated values have a heavier tail than equation (10).

On these assumptions a compound calculated at 53 K has a median true $T_c$ of 41, 36, 26 and 18 K at $\sigma$ = 0.15, 0.2, 0.3 and 0.4. In a pool of a million compounds of which 4.3% are calculated above 10 K, the most probable highest true $T_c$ is 50, 48, 42 and 34 K. The last two rows of ratios depend on the exponential form assumed for the true values, which the four compounds above 30 K cannot check.

For compounds predicted before they were made, the record cannot yet test the calculation. We have one such comparison at matched pressure for a compound calculated above 10 K. The highest onsets of Mg₂RhH₆, 24 K near 30 GPa and 29 K at 53 GPa, are a factor of 2.1 to 2.5 below the roughly 60 K its makers calculated for 30 to 50 GPa, in samples whose hydrogen content was inferred (the section on one atmosphere) [23]. Four intermetallics with an identified phase, whose calculations after screening gave 1.9 to 9.3 K, were measured lower by factors of 1.2 to 3.4 [16, 17]. The introduction of the second source prints values that would give 4.2. For $\sigma$ from 0.15 to 0.4 equation (12) gives ratios of 1.01 to 1.50 there, so selection as modelled does not account for the larger of the measured factors.

Correcting for selection means ranking on the posterior of the true value given the calculated one [110], which needs $\sigma$ and the distribution of true values in the candidate's own population. Neither is known beyond 50 meV per atom, and we give no corrected value for the compounds there.

One atmosphere

The third condition is persistence at one atmosphere. No hydride predicted to superconduct above 60 K at ambient pressure has been made at, or recovered to, one atmosphere. Table 9 lists eleven such predictions. Two did not form when their metals were heated in hydrogen at the right ratio, two are unstable in the one path-integral simulation run on them, one stayed intact in a short simulation with classical nuclei, and for six we found no test.

Outcomes for the predicted candidates

Figure 5 places the candidates against the scale of known metastability. Half of the known inorganic crystalline phases are metastable, with a median energy above the ground state of 15 meV per atom and a 90th percentile of 67 meV per atom [114].

Mg₂IrH₆. Of the two papers that predicted this compound, one placed it on the hull and the other described it as metastable [22, 64]. Syntheses up to 28 GPa and 2500 K, and a low-pressure autoclave route, produced Mg₂IrH₅ each time. Mg₂IrH₅ has the same metal sublattice with five sixths of the hydrogen sites filled and is a charge-balanced insulator [35]. The hydrogen-rich neighbour Mg₂IrH₇ forms above about 40 GPa, is also an insulator, and on decompression at room temperature reverts to Mg₂IrH₅ near 20 GPa without passing through the metallic composition between them [117]. Mg₂IrH₆ has not been observed, so whether it would last if made by another route is untested.

Li₂AuH₆ and Li₂AgH₆. They have the two highest calculated $T_c$ in the hull figure of the largest ambient-pressure survey, 88 and 111 K [5], and Li₂AuH₆ had passed harmonic and anharmonic phonon checks. In path-integral molecular dynamics at 80 K about a quarter of its hydrogen pairs into H₂ molecules and the rest diffuses. Superconductivity recalculated in that state gives 22 K, and the Hopfield sum we form from it is a factor of 4.2 below the linear-response value for the crystal [5, 37]. Li₂AgH₆ collapses in the same simulation [37]. The evidence is one study with density-functional forces, on 72 atoms for 1 ps and 243 atoms for 6 ps. We found no synthesis attempt for either compound.

The other predictions. Mg₂PtH₆ did not form in a study that explored the Mg-Pt-H system and did not name it as a target. Mg₃Pt and 2:1 Mg-Pt mixtures were heated in hydrogen at 160 bar and at 9 to 24 GPa, and all six high-pressure runs gave Mg₄Pt₃H₆ [36]. Li₂CuH₆ stayed intact in one molecular-dynamics run with classical nuclei [38], a weak test, since Li₂AuH₆ also stayed solid with classical nuclei at 80 K [37]. For the other six rows above 60 K we found no synthesis aimed at the compound and no simulation with moving nuclei. By our count from its figures, which carry no compositions, the largest survey holds 42 compounds of all chemistries calculated at or above 60 K [5].

Mg₂RhH₆. The calculations cited by its synthesis paper put this compound at 45 to 59 K, below the 60 K line. It is the one member of the family that has been made, by one group [23]. Nine cells gave resistive onsets between 18 and 29 K, with no magnetic measurement. The paper quotes 24 K near 30 GPa and 29 K at 53 GPa, against about 60 K that the same authors calculate for 30 to 50 GPa, a ratio of 2.1 to 2.5. The hydrogen content was inferred from the cell volume and one Raman mode, and the authors attribute part of the shortfall to hydrogen deficiency. On decompression at room temperature the resistance turned insulating at 28 GPa, and diffraction at 0.7 GPa showed Mg₂RhH₅. Release at low temperature has not been reported. Mg₂RhH₆ is the only compound in Table 9 that was made and then lost.

A low-pressure case: BaSiH₈. BaSiH₈ was never predicted at one atmosphere. Calculations place it on the hull above 130 GPa and find it dynamically stable down to 3 GPa in an anharmonic estimate, 5 GPa harmonically and 20 GPa with quantum nuclei, with a $T_c$ near 71 K at 5 GPa, and protected against loss of H₂ only above about 30 GPa [118, 119]. In 2026 a cubic Ba-Si-H phase was made by laser heating at 18 and at 31 GPa, and its diffraction pattern persists to ambient pressure [120]. Samples in that study are semiconducting or poorly metallic below about 40 to 50 GPa, and the only superconducting transition reported is 9 K at 142 GPa, where the authors' harmonic estimate is 52 K. The authors assign the phase as BaSiH₈, propose that its hydrogen has converted to molecular or Si-H covalent units, and note that BaSiH₆ and BaSiH₈ have nearly the same cell volume, so diffraction does not fix the hydrogen content. If the phase is BaSiH₈, its lattice outlasted the calculated limit of retention by 30 GPa. If it is not, the prediction is untested.

We tabulated the decompression outcomes of about twenty hydrides made under pressure, from papers published between 2020 and 2026; the list is ours and is not a census. Those recovered to ambient conditions include Y₃Fe₄H₂₀, U₄H₁₅, PdH₁.₃, seven lanthanide trihydrides, Mg₄Pt₃H₆ and the cubic Ba-Si-H lattice [36, 120, 121, 122, 123, 124]. Those lost on release include Mg₂IrH₇, Mg₂RhH₆, UH₇ and PdH₃ [23, 117, 122, 123], and we found no report of a clathrate superhydride recovered to one atmosphere. Within the list the highest measured $T_c$ is 2.9 K, in Mg₄Pt₃H₆, among compounds recovered at room temperature and 11.6 K, in ZrH₃, among those kept only cold [125]. Hydrides loaded by ion implantation or electrochemistry are outside the list. Among them PdH superconducts at 8 to 9 K (Table 3), and a palladium-copper alloy, implanted with hydrogen at liquid-helium temperature, at 16.6 K [126]. The hydride Th₄H₁₅ superconducts at about 8 K [127].

Stability and $T_c$ trade against each other

In the largest survey the best calculated $T_c$ at a given distance from the hull rises from 23 K on the hull to 111 K at 319 meV per atom, about 0.3 K for each meV per atom, and that front is a record of the best found (the section on screening) [5]. Among the stable cubic hydrides of the GNoME database the highest calculated $T_c$ is 23.5 K by the Allen-Dynes formula and 17 K after refinement with a first-principles Coulomb interaction, and the authors write that values of order 100 K appear systematically linked to thermodynamic instability [109]. A 2026 survey of about two million hydride structures finds more than 600 above 20 K, at roughly 100 meV per atom or more from the hull [128].

A 2025 calculation places fluorite-type BaRhH₈ on the ambient-pressure hull with a calculated $T_c$ of 52 K, and AcRhH₈ 6 meV per atom above it with 78 K [115]. Its hull uses a different set of competing phases from the surveys above, and neither compound has been tested beyond harmonic phonons. If the calculation stands, the two are exceptions to the trade.

For the six rows of Table 9 that give a first published value, the hull distance rose by about 40 to 190 meV per atom by October 2026. The papers do not say which competing phases moved them, but the database lists the decomposition products [96]. For Mg₂IrH₆ and Mg₂RhH₆ they are the compositions with five and seven hydrogen atoms. For Mg₂PdH₆, KInH₃ and Li₂CuH₆ they include MgH₆, KH₇ and CuH₄, which we do not know to have been made at one atmosphere, so part of the rise for those three may not correspond to compounds that exist. In the Matbench Discovery benchmark about 33,000 structures lie below the Materials Project hull, and about 20,000 of them remain on the hull once the benchmark's own structures are added [129].

That strong coupling at the Fermi level drives phonons toward instability is known [75], and two hydride surveys report that high calculated $T_c$ goes with distance from the hull [63, 109]. We put the link in the terms of this paper, as a hypothesis. A large $\eta_H$ means that the energies of electrons at the Fermi level change steeply when hydrogen moves. A lattice can use the same sensitivity to lower its energy, by rearranging its hydrogen at fixed composition or by taking up or releasing hydrogen until the electron count closes a shell. The Mg₂IrH₆ family shows the second route. Its metallic composition sits between two insulators one hydrogen atom away on either side, Mg₂IrH₅ with a calculated gap of 0.65 eV and Mg₂IrH₇ with 2.46 eV [35, 117]. Li₂AuH₆ shows the first in simulation, where pairing a quarter of its hydrogen into molecules removes three quarters of its Hopfield sum. In the Fröhlich model with a structureless coupling, the conduction electrons lower the force constant of a mode by twice its Hopfield parameter [29, 75]. The bare force constant is not observable in a real metal, so we take from the relation only that the most strongly coupled modes are the ones at risk. Li₂AuH₆, which has about 70% of its $\lambda$ below 30 meV [68] and fails once its nuclei move, fits this account. One case does not test it.

Pressure opposes these instabilities through the $PV$ term in the enthalpy. A phase that is denser by one cubic ångström per atom is favoured by 624 meV per atom at 100 GPa, nine times the 90th-percentile metastability of known compounds, and it loses all of that on release. In the superhydrides pressure therefore stabilises a hydrogen-rich metal against the insulating phases that compete with it, and it raises the hydrogen density and with it $\eta_H$.

Hull distance and phonon stability do not predict survival

The standard computational filters are energy above the hull and the absence of imaginary phonon frequencies. That neither is a criterion for lifetime is known. The authors of the metastability scale present it as a description of observed phases and leave kinetics to later work [114], and the prediction of BaSiH₈ added a kinetic threshold for this reason [118].

Mg₂RhH₆ passed both filters and was lost on decompression. Li₂AuH₆ passed the phonon test, with and without anharmonic corrections, and is unstable in path-integral dynamics. The cubic Ba-Si-H lattice is recovered below both limits calculated for BaSiH₈. The high-pressure phases of Bi₀.₅Sb₁.₅Te₃ fail the phonon test at zero pressure, and superconductivity was retained there after a quench [130]. Aluminium hydride lies 120 meV per atom above aluminium and hydrogen gas in Gibbs energy at room temperature and is stable only above about 7 kbar of hydrogen, yet it keeps at ambient conditions because its hydrogen release is limited by kinetics [131].

The barrier needed for a lifetime of one year at 300 K depends on which step limits the loss. If each hydrogen site empties independently by one activated event with an attempt frequency of $10^{13}$ per second, the barrier must be 1.22 eV, or 1.0 eV at $10^{10}$ per second; a week at 200 K needs 0.75 eV. If one such event anywhere among $N$ sites starts a transformation that then runs quickly, the requirement rises by $k_BT\ln N$, to 1.9 eV for $N = 10^{12}$, the number of hydrogen atoms in a grain about 2 μm across. If loss is limited by diffusion to a free surface, a plate of thickness $L$ empties in a time $L^2/\pi^2D$, and with a diffusion prefactor of $10^{-3}$ cm² per second the migration barrier must be 0.68 eV at 10 μm, 0.45 eV at 1 mm and 0.33 eV at 1 cm.

The calculated activation energy for hydrogen migration is 0.23 eV in palladium and 0.46 eV for hydride ions in the polyhydride CaH₄ under pressure, the second from molecular dynamics at 1000 K and above [132, 133]. With the same prefactor a 10 μm grain empties in about a second at 0.23 eV and in an hour and a half at 0.46 eV, so bulk diffusion does not hold hydrogen in grains of the size made in diamond anvil cells. At 0.46 eV a plate 1 mm thick has a relaxation time of 1.7 years, so slow diffusion could contribute to retention in a bulk sample. In a grain the hydrogen has to be held by a barrier to nucleating the hydrogen-poor phase, by a barrier to recombination at the surface, or by strong binding in a closed-shell unit. The complex hydrides Mg₂IrH₅ and Mg₄Pt₃H₆ have the last of these, and neither superconducts above 3 K [35, 36].

A decomposition temperature can be read as a barrier through the per-site formula, $E = k_BT\ln(\nu t)$ for loss in a time $t$. The result is a scale that assumes one rate-limiting step and moves by 0.04 to 0.08 eV for each factor of ten in the assumed prefactor $\nu$. With $\nu = 10^{13}$ per second, ZrH₃, which decomposes to ZrH₂ between 200 and 270 K when heated at 10 K per minute in vacuum, reads 0.6 to 0.85 eV [125]. Y₃Fe₄H₂₀, unchanged after 30 hours in air at room temperature and 7 to 8% smaller in volume after one to three months, reads 1.1 to 1.2 eV [121]. AlH₃ is the one hydride here with measured kinetics. It decomposes between 60 and 140 °C by nucleation and growth, with an activated complex of about nine formula units and an activation energy of about 1.06 eV [131, 134], where the per-site reading of the same decomposition times is about 1.2 eV. Isothermal decomposition curves from the same laboratory [135], as we read them, correspond to an effective prefactor near $10^{10}$ per second and extrapolate to about a third of the hydrogen lost in roughly a year at 300 K. That report describes fresh material as pyrophoric and surface-treated material as stored at ambient conditions for more than twenty years. In the one case with known kinetics the lifetime depends on nucleation and on the surface as well as on a barrier.

For the compounds of Table 9 we found no published rate, single-event barrier or lifetime for hydrogen loss. The only path calculations we found are nudged-elastic-band results for Mg₂IrH₆, which give no barrier along one path for inserting hydrogen into Mg₂IrH₅ and 32 meV per atom between two Mg₂IrH₆ polymorphs [35].

States retained at ambient pressure

Cooling under pressure and releasing cold is an established way to recover binary transition-metal hydrides made at several GPa, and it is how ZrH₃ is kept [125, 136]. Deng, Chu and coworkers applied cold release to pressure-enhanced superconductors in diamond anvil cells. In FeSe a quench from 4 GPa at 4.2 K retains an onset of 37 K against 9 K for the ambient phase. That state is gone after warming to about 200 K, and heating to 300 K leaves a strained crystal with a $T_c$ near 20 K [137]. On the per-site scale a state lost near 200 K within minutes to a week reads 0.6 to 0.75 eV at $10^{13}$ per second. For prefactors from $10^{6}$ to $10^{13}$ per second the reading stays between 0.4 and 0.6 of the requirement for a year at 300 K.

In HgBa₂Ca₂Cu₃O₈₊δ quenched from 10 to 30 GPa, five samples retained resistive onsets of 139 to 151 K, against 133 to 135 K for the equilibrium phase. Zero resistance after the quench is not reported. Cycling to 170 K lowered the onset slightly, cycling to room temperature lowered one sample from 147 to 143 K, and a sample retrieved from the cell through room temperature showed about 140 K with a shielding fraction of about 78% in magnetisation. Part of the enhancement therefore outlasts room temperature for the time of handling, and a loss this gradual is not described by one barrier. Diffraction finds the same tetragonal structure with broader peaks, and the authors attribute the retention to strain or defects [10]. The quench results on superconductors come from one group, and we found no independent replication.

Epitaxial strain substitutes for pressure in films a few unit cells thick. Bulk La₃Ni₂O₇ shows signatures of superconductivity near 80 K only above about 14 GPa [138]. Compressively strained La₃Ni₂O₇ films superconduct at ambient pressure [139], and (La,Pr)₃Ni₂O₇ films reach an onset of 63 K and zero resistance at 37 K [140].

The boron-carbon clathrate SrB₃C₃ was recovered from about 50 GPa to one atmosphere, where it persists in an inert atmosphere and degrades in moist air within hours [141]. Its superconducting onset is about 20 K at 40 GPa, and we found no measurement of its transition at one atmosphere [142]. Across chemistries the accessible range of metastability scales with cohesive energy [114].

Data on persistence

Electron-phonon calculations have been published for more than ten thousand metals at ambient pressure [5]. The record of persistence has not been assembled. Beyond the twenty or so outcomes we tabulated, older records exist that we have not collected. Binary hydrides of the 3d and 4d metals of groups VI to VIII have been made at several GPa, cooled under pressure and studied at one atmosphere [136]. RhH₂, made at 8 GPa and released cold, keeps its hydrogen indefinitely at 77 K and for minutes at 150 K [143]. Room-temperature equilibrium pressures have been compiled for about five thousand storage-hydride compositions; they describe stable phases and do not record whether a metastable hydride survives [144]. We found no compilation of recovery outcomes with their protocols, and no test of a descriptor against one.

We found calculated barriers along a transformation path for two hydrides, BaSiH₈ and Mg₂IrH₆, and one path-integral simulation that tests whether an ambient-pressure candidate survives [35, 37, 118]. A model that ranks candidates by a predicted $T_c$ and filters on hull distance and harmonic phonons ranks on the quantity for which data are plentiful and filters with two tests that have each failed in the cases above. For the four candidates above 60 K that did not form or were unstable in simulation, the outcome was decided by which phase forms or by the motion of hydrogen, and we found no labelled set for either.

When phonons do not provide the pairing

The ambient-pressure record belongs to the cuprates, which are classed as unconventional superconductors and whose pairing mechanism is not settled [138, 145]. The iron-based superconductors reach 55 to 56 K in bulk [146, 147]. A review of single-layer FeSe on SrTiO₃ quotes a $T_c$ of 65 K, and 109 K from a single in situ transport measurement, against 9 K for bulk FeSe, and weighs phonon pairing, magnetic or orbital pairing, and the two acting together [148, 149]. This paper treats the two kinds of pairing separately and does not cover a mixed case.

The cuprate stiffness scale is close to $T_c$

From equation (2) Carlson and coauthors tabulate a $T_\theta$ of 130 to 190 K for the mercury compound, whose $T_c$ is 133 to 135 K, 140 K against a $T_c$ of 85.5 K for optimally doped Y₀.₈Ca₀.₂Ba₂Cu₃O₇₋δ, and 54 K against 38 K for optimally doped La₂₋ₓSrₓCuO₄. The ratios are 1.0 to 1.6, where MgB₂ has 36 [25]. The authors caution that the prefactor depends on microscopic details, so these ratios suggest that stiffness limits $T_c$ in the cuprates and do not establish it. In underdoped cuprates $T_c$ rises linearly with superfluid density over part of the doping range [150].

Putting $T_\theta$ at 300 K in equation (2) requires an in-plane penetration depth of 106 nm or less at a layer spacing of 6 Å. The optimally doped cuprates in the same tabulation have 150 to 270 nm. For the mercury cuprate the requirement is 1.6 to 2.3 times its tabulated stiffness.

Uemura's empirical upper limit on $T_c/T_F$, about 0.05 with $T_F$ derived from the superfluid density, describes the cuprates and the other strongly correlated superconductors in his compilation [151]. Gate-doped LiₓZrNCl, a two-dimensional superconductor, reaches $T_{\text{BKT}}/T_F = 0.12$ at low carrier density [152], close to the bound of 1/8 for a single parabolic band in two dimensions [40]. At a ratio of 0.05 a 300 K transition needs a Fermi temperature of 6000 K, which is 1.3 carriers per CuO₂ cell if the effective mass is four electron masses, the value we assume for a cuprate-like band. At 0.12 it needs 2500 K and about half a carrier per cell.

Calculated $T_c$ is a few percent of the electronic energy scales

Qin and Yang find $T_c/J$ peaking between 0.04 and 0.07 across several effective pairing models, with the best cuprates at 0.06 to 0.067, and conclude that 300 K at ambient pressure would need a superexchange $J$ of 400 to 700 meV [153]. The cuprates in their table have 105 to 176 meV. The dynamical cluster result $T_c = 0.023\,t$ for the Hubbard model at $U = 4t$ and 10% doping [154] would need a hopping $t$ near 1.1 eV, against the 0.4 eV we take for the bare one-band hopping of a cuprate. Both are numerical results on simplified models, and Qin and Yang say that theirs carries no proof. On these two results a cuprate-like route to 300 K needs $J$ at 2.3 to 4.0 times the largest value in that table, or $t$ at 2.8 times the value we take.

No validated forward model

We know of no controlled, validated theory that returns $T_c$ from a crystal structure in this class. A dynamical vertex calculation on a one-band model of NdNiO₂ gave the doping range of the infinite-layer nickelates three weeks before the measured dome was posted, with a maximum $T_c$ of 36 K against the 15 K then measured at 20% strontium; changing one interaction parameter from $8t$ to $9t$ nearly removes the difference [155, 156]. A density-matrix embedding calculation from the crystal structure returns $d$-wave order and a pairing gap for CaCuO₂ at three pressures and for two mercury cuprates. Its $T_c$ is a weak-coupling conversion of the gap, about 180 K for CaCuO₂ at ambient pressure against about 110 K measured in the strontium-substituted compound [157]. We found no blind test of either method on a second family. A 2022 review of the numerical methods calls the mapping from a material to model parameters neither systematic nor controlled [158].

Ground-state studies of the Hubbard and $t$-$t'$-$J$ models find stripes with no superconducting order in the pure Hubbard model at 1/8 doping [159] and disagree on whether next-nearest-neighbour hopping gives pairing on the hole-doped side [160, 161]. None of the three reports a $T_c$.

Four material-specific descriptors have been correlated with the maximum $T_c$ within the cuprates: the charge-transfer energy, the hopping range that grows with the apical oxygen distance, the superexchange and the hole content on planar oxygen [162, 163, 164, 165]. They are not independent, since the superexchange depends on the charge-transfer energy $E$ as $J\approx4t^4/E^3$ [166]. The published correlations are in-sample trends across eight to sixteen compounds or families [162, 164, 165], and we found no test of any descriptor on compounds held out of the fit. La₂₋ₓSrₓCuO₄ has 1.5 times the $J$ of YBa₂Cu₄O₈ and half its $T_c$ [153].

A nickel analogue of the cuprates was proposed in 1999 and found in 2019, at 9 to 15 K [167, 168]. La₃Ni₂O₇ was named in 2017 as a possible realisation of a bilayer model that might exceed cuprate $T_c$, with the note that it did not superconduct. Signatures of superconductivity near 80 K were reported in 2023 between 14 and 43.5 GPa, and the proposal did not mention pressure [138, 169]. Norman wrote in 2016 that the heavy-fermion, cuprate and iron-based superconductors each took the community by surprise [145].

What a model can do in this class

From a band structure a model can compute an upper bound on the superfluid stiffness of a hypothetical layered compound, because the stiffness cannot exceed the optical sum-rule weight of the bands [40]. Inside a known family it can rank compounds on the descriptors above, which have been tested only in sample. The record gives no basis for proposing a new family, and we do not ask roomtsc to try.

The pressure-quenched mercury cuprate, with an onset up to 151 K, and compressively strained nickelate films, with an onset of 63 K, each show at ambient pressure a $T_c$ that the bulk equilibrium material reaches only under pressure [10, 140]. Both are treated under "States retained at ambient pressure". The strain in the films is held by their substrate.

What roomtsc estimates

roomtsc is a staged estimator, and none of it has been trained. For a candidate crystal structure at one atmosphere it returns an estimate of each condition with an uncertainty, and a decision layer uses those estimates to choose which calculation or experiment to run next. Table 10 lists the components.